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Task execution of multiagent systems in social networks (MAS-SN) can be described through agents' operations when accessing necessary resources distributed in the social networks; thus, task allocation can be implemented based on the agents' access to the resources required for each task and aimed to minimize this resource access time. Currently, in undependable MAS-SN, there are deceptive agents that may fabricate their resource status information during task allocation but not really contribute resources to task execution; although there are some game theory-based solutions for undependable MAS, but which do not consider minimizing resource access time that is crucial to the performance of task execution in social networks. To achieve dependable resources with the least access time to execute tasks in undependable MAS-SN, this paper presents a novel task allocation model based on the negotiation reputation mechanism, where an agent's past behaviors in the resource negotiation of task execution can influence its probability to be allocated new tasks in the future. In this model, the agent that contributes more dependable resources with less access time during task execution is rewarded with a higher negotiation reputation, and may receive preferential allocation of new tasks. Through experiments, we determine that our task allocation model is superior to the traditional resources-based allocation approaches and game theory-based allocation approaches in terms of both the task allocation success rate and task execution time and that it usually performs close to the ideal approach (in which deceptive agents are fully detected) in terms of task execution time.